We show that the six-vertex model with parameter $c\in[\sqrt 3, 2]$ on a square lattice torus has an ergodic infinite-volume limit as the size of the torus grows to infinity. Moreover we prove that for $c\in[\sqrt{2+\sqrt 2}, 2]$, the associated height function on $\mathbb Z^2$ has unbounded variance.
The proof relies on an extension of the Baxter-Kelland-Wu representation of the six-vertex model to multi-point correlation functions of the associated spin model. Other crucial ingredients are the uniqueness and percolation properties of the critical random cluster measure for $q\in[1,4]$, and recent results relating the decay of correlations in the spin model with the delocalization of the height function.
Delocalization of the height function of the six-vertex model
Institut des Hautes Études Scientifiques（高等科学研究所）；Université de Genève（日内瓦大学）；Åbo Akademi University（奥布皇家学院）；Département de mathématiques, Université de Fribourg（弗里堡大学数学系）；Université Paris-Saclay（巴黎萨克雷大学）
Graph homomorphisms from the $\mathbb{Z}^d$ lattice to $\mathbb{Z}$ are functions on $\mathbb{Z}^d$ whose gradients equal one in absolute value. These functions are the height functions corresponding to proper $3$-colorings of $\mathbb{Z}^d$ and, in two dimensions, corresponding to the $6$-vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures. Additional results are obtained in higher dimensi…
We consider a class of probability distributions on the six-vertex model, which originate from the higher spin vertex models in arXiv:1601.05770 and have previously been investigated in arXiv:1610.06893. For these random six-vertex models we show that the asymptotic behavior near their base is asymptotically described by the GUE-corners process.
We prove that Schütz's ASEP Markov duality functional is also a Markov duality functional for the stochastic six vertex model. We introduce a new method that uses induction on the number of particles to prove the Markov duality.
In this note we establish the convergence of the stochastic six-vertex model to the one-dimensional asymmetric simple exclusion process, under a certain limit regime recently predicted by Borodin-Corwin-Gorin. This convergence holds for arbitrary initial data.
We prove an identity that relates the q-Laplace transform of the height function of a (higher spin inhomogeneous) stochastic six vertex model in a quadrant on one side, and a multiplicative functional of a Macdonald measure on the other. The identity is used to prove the GUE Tracy-Widom asymptotics for two instances of the stochastic six vertex model via asymptotics analysis of the corresponding Schur measures.
We construct an irreversible local Markov dynamics on configurations of up-right paths on a discrete two-dimensional torus, that preserves the Gibbs measures for the six vertex model. An additional feature of the dynamics is a conjecturally nontrivial drift of the height function.
We consider the inhomogeneous stochastic six vertex model with periodicity starting from step initial data. We prove that it converges almost surely to a deterministic limit shape. For the proof, we map the stochastic six vertex model to a deformed version of the discrete Hammersley process. Then we construct a colored version of the model and apply Liggett's superadditive ergodic theorem. The construction of the colored model includes a new idea using a Boolean-type product, which generalizes and simplifies the method used in arXiv:2204.11158.
We consider the stochastic higher spin six vertex (SHS6V) model introduced in [Corwin-Petrov, 2016] with general integer spin parameters $I, J$. Starting from near stationary initial condition, we prove that the SHS6V model converges to the KPZ equation under weakly asymmetric scaling. This generalizes the result of [Corwin-Ghosal-Shen-Tsai, 2018] from $I = J =1$ to general $I, J$.
In this paper, we prove that the stochastic telegraph equation arises as a scaling limit of the stochastic higher spin six vertex (SHS6V) model with general spin $I/2, J/2$. This extends results of Borodin and Gorin which focused on the $I=J=1$ six vertex case and demonstrates the universality of the stochastic telegraph equation in this context. We also provide a functional extension of the central limit theorem obtained in [Borodin and Gorin 2019, Theorem 6.1]. The main idea is to generalize the four point relation established in [Borodin and Gorin 2019, Theorem 3.1], using fusion.
We prove a positivity theorem for a certain family of operators defined in terms of the stochastic six-vertex model. We explore connections of this result with other vertex models and $t$-deformed Schensted insertions.
In this paper we consider the existence and uniqueness of pure states with some fixed slope $(s, t) \in [0, 1]^2$ for a general ferroelectric six-vertex model. First, we show there is an open subset $\mathfrak{H} \subset [0, 1]^2$, which is parameterized by the region between two explicit hyperbolas, such that there is no pure state for the ferroelectric six-vertex model of any slope $(s, t) \in \mathfrak{H}$. Second, we show that there is a unique pure state for this model of any slope $(s, t)$ on the boundary $\partial \mathfrak{H}$ of $\mathfrak{H}$. These results confirm predictions of Buk…
In this paper we consider the stochastic six-vertex model on a cylinder with arbitrary initial data. First, we show that it exhibits a limit shape in the thermodynamic limit, whose density profile is given by the entropy solution to an explicit, non-linear conservation law that was predicted by Gwa-Spohn in 1992 and by Reshetikhin-Sridhar in 2018. Then, we show that the local statistics of this model around any continuity point of its limit shape are given by an infinite-volume, translation-invariant Gibbs measure of the appropriate slope.
From logarithmic delocalization of the six-vertex height function under sloped boundary conditions to weakened crossing probability estimates for the Ashkin-Teller, generalized random-cluster, and $(q_σ,q_τ)$-cubic models
In this paper we consider two models in the Kardar-Parisi-Zhang (KPZ) universality class, the asymmetric simple exclusion process (ASEP) and the stochastic six-vertex model. We introduce a new class of initial data (which we call {\itshape generalized step Bernoulli initial data}) for both of these models that generalizes the step Bernoulli initial data studied in a number of recent works on the ASEP. Under this class of initial data, we analyze the current fluctuations of both the ASEP and stochastic six-vertex model and establish the existence of a phase transition along a characteristic lin…
We apply the Tangent Method of Colomo and Sportiello to predict the arctic curves of the Six Vertex model with reflecting (U-turn) boundary and of the related Twenty Vertex model with suitable domain wall boundary conditions on a quadrangle, both in their Disordered phase.
A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six vertex model in a quadrant. The corresponding law of large numbers -- the limit shape of the height function -- is described by the (determinis…
In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time $T$, it is known that the one-point height function fluctuations are of order $T^{1/3}$ and governed by the Tracy-Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $T^{2/3}$ and vertically by $T^{1/3}$, converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin-Bufetov-Wheeler between the stochastic six-vertex model and the ascending…
In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $Δ<1$. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches $1/2$. This latter result is at the base of a number of recent results, in particular the rigorous pro…
We show that limit shapes for the stochastic 6-vertex model on a cylinder with the uniform boundary state on one end are solutions to the Burger type equation. Solutions to these equations are studied for step initial conditions. When the circumference of the cylinder goes to infinity the solution corresponding to critical initial densities coincides with the one found by Borodin, Corwin and Gorin.
In this paper we consider the Higher Spin Six Vertex Model on the lattice $\mathbb{Z}_{\geq 2} \times \mathbb{Z}_{\geq 1}$. We first identify a family of translation invariant measures and subsequently we study the one point distribution of the height function for the model with certain random boundary conditions. Exact formulas we obtain prove to be useful in order to establish the asymptotic of the height distribution in the long space-time limit for the stationary Higher Spin Six Vertex Model. In particular, along the characteristic line we recover Baik-Rains fluctuations with size of chara…
We introduce a family of Markov growth processes on discrete height functions defined on the 2-dimensional square lattice. Each height function corresponds to a configuration of the six vertex model on the infinite square lattice. We focus on the stochastic six vertex model corresponding to a particular two-parameter family of weights within the ferroelectric regime. It is believed (and partially proven, see Aggarwal, arXiv:2004.13272) that the stochastic six vertex model displays nontrivial pure (i.e., translation invariant and ergodic) Gibbs states of two types, KPZ and liquid. These phases …
Our results in this paper are two-fold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time $T$, and show that they are of order $T^{1 / 3}$ along a characteristic line. Upon scaling by $T^{1 / 3}$, we establish that these fluctuations converge to the long-time height fluctuations of the stationary KPZ equation, that is, to the Baik-Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and…
We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In part…
We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a fam…
We consider the ASEP and the stochastic six vertex model started with step initial data. After a long time, $T$, it is known that the one-point height function fluctuations for these systems are of order $T^{1/3}$. We prove the KPZ prediction of $T^{2/3}$ scaling in space. Namely, we prove tightness (and Brownian absolute continuity of all subsequential limits) as $T$ goes to infinity of the height function with spatial coordinate scaled by $T^{2/3}$ and fluctuations scaled by $T^{1/3}$. The starting point for proving these results is a connection discovered recently by Borodin-Bufetov-Wheeler…
The six-vertex model in statistical physics is a weighted generalization of the ice model on $\mathbb{Z}^2$ (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model depicts its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require…
Gaussian free field convergence of the six-vertex model with $-1\leqΔ\leq-\frac12$
Institut des Hautes Études Scientifiques（高等科学研究所）；Université de Genève（日内瓦大学）；ENSL, CNRS, Laboratoire de Physique（里昂高等师范学院，法国国家科学研究中心，物理实验室）；CNRS, Sorbonne Université, LPSM（法国国家科学研究中心，索邦大学，概率论与统计数学实验室）；Université de Fribourg（弗里堡大学）
We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions. For free-fermion vertex weights the partition function can be expressed in terms of some Hankel determinant, or equivalently as a Coulomb gas with discrete measure and a non-polynomial potential with two hard walls. We use Coulomb gas methods to study the partition function in the thermodynamic limit. We obtain the free energy of the six-vertex model as a function of the parameters describing the geometry of the scaled L-shaped domain. Under variations of these parameters the sys…
We prove that the joint distribution of the values of the height function for the stochastic six vertex model in a quadrant along a down-right path coincides with that for the lengths of the first columns of partitions distributed according to certain Hall-Littlewood processes. In the limit when one of the quadrant axes becomes continuous, we also show that the two-dimensional random field of the height function values has the same distribution as the lengths of the first columns of partitions from certain ascending Hall-Littlewood processes evolving under a Robinson-Schensted-Knuth type Marko…
We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter $Δ\to 1^+$ so as to zoom into the ferroelectric/disordered phase critical point) its height function fluctuations converge to the solution to the KPZ equation. We also prove that the one-dimensional family of stochastic Gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the stochastic Burgers equation. Our proofs rely upon the Markov (self) duality of our model. The starting point is an exact microscopic Hopf-Cole transform fo…
In this paper we consider a class of probability distributions on the six-vertex model from statistical mechanics, which originate from the higher spin vertex models of https://arxiv.org/abs/1601.05770. We define operators, inspired by the Macdonald difference operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. The development of our operators is largely based on the properties of a remarkable family of symmetric rational functions, which were previously studied in https://arxiv.org/abs/1410.0976. For the class of model…
In this article we study the stochastic six vertex model under the scaling proposed by Borodin and Gorin (2018), where the weights of corner-shape vertices are tuned to zero, and prove Conjecture 6.1 therein: that the height fluctuation converges in finite dimensional distributions to the solution of stochastic telegraph equation.
We present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the Macdonald processes hierarchy. First, we show how Macdonald $q$-difference operators with $t=0$ (an algebraic tool crucial for studying the corresponding Macdonald processes) can be utilized to get $q$-moments of the height function $\mathfrak{h}$ in the higher spin six vertex model first computed in arXiv:1601.05770 using Bethe ansatz. This result in particular implies that for the vertex model with the step Bernoulli boundary condition, the…
The six-vertex model is an important model in statistical physics and has deep connections with counting problems. There have been some fully polynomial randomized approximation schemes (FPRAS) for the six-vertex model [30, 10], which all require that the constraint functions are windable. In the present paper, we give an FPRAS for the six-vertex model with an unwindable constraint function by Markov Chain Monte Carlo method (MCMC). Different from [10], we use the Glauber dynamics to design the Markov Chain depending on a circuit decomposition of the underlying graph. Moreover, we prove the ra…
We consider a homogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a family of s…
Square ice is a statistical mechanics model for two-dimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a so-called 6-vertex model configuration, and argue that its scaling limit is a space-filling version of the random fractal curve SLE$_κ$, Schramm--Loewner evolution with parameter $κ$, where $4<κ\leq 12+8\sqrt{2}$. For square ice, $κ=12$. At the "free-fermion point" of the 6-vertex model, $κ=8+4\sqrt{3}$. These unusual values lie outside the classical interval $2\le κ\…
